Time-Ordered Perturbations

Gilbert A. Ibitola, Olanrewaju Ajanaku, Abiola Olawale Ilori

There exist several forms of perturbation theory in Quantum Mechanics, namely: non-degenerate perturbation theory, degenerate perturbation theory, time-dependent perturbation theory, time-constant perturbation theory and time-harmonic perturbation theory. This paper presents an extension of the perturbation theory for non-degenerate states. It has been observed that most texts and journal papers treat only the first –order and second-order non-degenerate perturbation theory. Therefore, this paper attempts to treat and present the third order and fourth order non-degenerate perturbation theory in quantum mechanics. It can thus be asserted that the higher the order of perturbations of quantum systems that we know, the more successful will be our efforts in suppressing or eliminating them.

Stanisław Olszewski

We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing N-1. The detailed calculations are performed for the perturbation energy of orders N=7 and N=8 .

Peter J. Rossky, Martin Karplus

By use of basic results of diagrammatic perturbation theory and combinatorial arguments, a formula is derived for the number of topologically distinct, connected, time ordered Goldstone diagrams contributing to the nth order perturbation energy. A general perturbation consisting of one- and two-particle operators is considered. The result is extended to an effective one-particle operator that implicitly includes all equal-time contractions and to multiple one-particle perturbations that enter only a restricted number of times in any order; the former is important for a Hartree–Fock zero order function and the latter for a system with external or internal field perturbations. It is pointed out that the method for computing the number of disconnected diagrams is quite general and independent of the nature of the perturbation and that the connected topologies which possess unusual symmetry (i.e., those for which all sets of vertex space–time labels are not distinct) are of a very specific and easily identified type. Numerical results for the various cases considered are given through 5th order.

John B. Ketterson, Shenian N. Song